A Naming System for Hyperoperator-related Functions up to f_ε0(n)

(or just "Hyperoperator Names")

Last updated 21/11/23

To skip the talk and see a full table of the function names, you can view the function name list.

Section 1: Introduction (A Game With an Issue)

Hyperoperators are a continuation of mathematical operators past the standard addition, multiplication and exponentiation. The most well known of these is tetration, although the list continues with pentation, hexation, and so on. These operators roughly correspond to n{a}m where {a} refers to a up arrows, or approximately f_a(n) in the fast-growing hierarchy, or {n,m,a} in array notation. The natural next step, if you are intent on surpassing the limit of the hyperoperator list, is to recurse on it. Doing so brings us to something like n{n}n, where the hyperoperator used increases as n increases. Such a function is equivalent to {n,n,n} in array notation, and roughly equivalent to f_ω(n) in FGH.

In 2021 I created Endless Stairwell, an incremental game designed mainly to utilise the full scale of ExpantaNum.js and hold the largest numbers ever used in a major game. Endless Stairwell caps out at around {10, 10¹⁵, 1, 2}, roughly equivalent to f_ω+1(10¹⁵) in FGH. I added this game to an existing online list of "incremental game limits", at which point I didn't have a name for the growth rate of the game's largest numbers. No problem, the googology wiki helpfully lists expansion as the function name for {a,b,1,2} ≈ f_ω+1(b). This name was defined by Jonathan Bowers in his page on the Exploding Array Function some time around 2008. Unfortunately however, no such name exists for f_ω(n). In this page I seek to create a continuous system of names to fill this gap and continue further.

Section 2: The Basic System

Since the first recursion on the existing named hyperoperators is at f_ω(n), it'd be fair to call it megation (from "omega"). I am aware that the word is already in use to refer to some other specific things, but they seem to be niche. I am also aware that it sounds kind of dumb. Doesn't mean I'm going to change it.

An exact definition for megation is as follows: a megated to b = {a,a,b} in array notation. This gives it the appropriate growth rate of f_ω(n). The prefix version of megation is mega-, which will be useful for larger growth rates. The next growth rate is expansion, or {a,b,1,2} ≈ f_ω+1(n). The name for this in my system is mega-addition. After this comes mega-multiplication ({a,b,2,2} ≈ f_ω+2(n)), then mega-exponentiation ({a,b,3,2} ≈ f_ω+3(n)), and so on. This too can be recursed on with duomegation (megation with the prefix duo-), which is equal to {a,a,b,2} ≈ f_ω2(n). Bowers defines {a,b,1,3} as "explosion", although in my system it is called duomega-addition. From here on let's make continuing the system easier to understand by breaking down the related ordinals in FGH.

Most major ordinals below ε₀ can be thought of as a series of decreasing segments x₁ + x₂ + x₃ ... where each segment x is either a more simple ordinal, with the exception of the last segment, which can sometimes be a finite number. an example of this is ω²+ω2+1, with the segments being ω², ω2 and 1. These segments, when simplified, can be thought of as being composed of a tetrational factor, an exponential factor and a multiplicative factor. For example, (ω³)2 can be said to have an exponential factor of 2 and a multiplicative factor of 1 (and a tetrational factor of 0) (Side note: this means that each segment could be displayed as an array for each factor, and thus every ordinal below ε₀ can be displayed using a 2-dimensional array^{[proof needed]}, which would be a weird notation that I might flesh out at some point). Each segment in the ordinal can be thought of as equivalent to a segment in my naming system. To take an existing example: Mega-addition (f_ω+1(n)) has two segments, the mega (ω) segment and the addition (1) segment, seperated by a hyphen. It can be said for simplicity that mega-addition has the "equivalent ordinal" ω+1.

Functions with equivalent ordinals of multiples of ω use the standard prefixes for numerals (duo-, tri-, quadri-, quinti-, sexti-, septi-, octi-, noni-, deci-, etc.), for example trimegation ({a,a,b,3}) has the equivalent ordinal ω3 and is thus rougly equivalent to f_ω3(n).

Here is a table for some names based on the system as defined so far:

The limit for this is {a,a,a,b} ≈ f_ω²(n). It is obvious where to go from here.

Section 3: Continuation up to f_ε0(n)

The next step after multiples of ω is expomegation (expo- from exponentiation), which is defined as {a,a,a,b} in array notation. This means it has a growth rate of f_ω²(n). Something important to note is that "duoexpomegation" is not f_(ω²)2(n), but instead f_ω³(n). expoduomegation is f_(ω²)2(n), and f_(ω³)2(n) would be duoexpoduomegation. This is where the use of segments starts to become more clearly useful; f_ω²+ω+1(n) would be expomega-mega-addition.

The final step towards reaching f_ε0(n) is to give names for tetrational (power-tower) ordinals such as ω^ω and ω^ωω. Although ordinal tetration is not well-defined, ordinals such as ω^ω can be informally simplified into ω↑↑2 (or ²ω), making it more clear that the ordinal has a tetrational factor. To refer back to the factors with another example: ω^(ω3)4 has a tetrational factor of 1, an exponential factor of 2, and a multiplicative factor of 3 (Although I'm pretty sure this doesn't make it equal to ((ω^^2)³)4, if that's even a valid calculable ordinal). These factors make it easy to create function names; tetramegation (tetra- from tetration) is defined as {a,b[2]2} in array notation (specifically BAN). It has a growth rate of f_ω^ω(n). Duotetramegation has a corresponding ordinal of ω^ωω and is defined as {a,b[1,2]2}.

UPDATE 21/11/23: For cases that do not fit the earlier criteria, issues may arise. For example, f_ω^ω+1(n) would be called "tetramega-addition", but such a name would also apply to f_ω^ω+1(n). Although my initial idea to solve this was to use brackets when necessary (So f_ω^ω+1(n) would be called "tetra(mega-addition)"), I think a nicer way would be to use "-by-" where brackets are required to differentiate two names. For example, this would give f_ω^ω+1(n) the name "tetra-by-mega-addition". f_ω^ω2(n) would be named "tetra-by-duomegation".

This system ends at f_ε0(n), the function for which I am going to give two different names: The special name Epsilonation (See section 4), and the extrapolated name Pentamegation (See section 5).

Section 4: Special Names for Functions Past f_ε0(n)

This one is a bit of a quick bonus, since these function names doesn't follow any sort of continuous system.

Epsilonation is defined as {a,b[[1]]2} = {a,b[1\2]2}. It has the equivalent ordinal of ε₀ and has a growth rate of f_ε0(n).
Zetation is defined as {a,b[1\1\2]2}. It has the equivalent ordinal of ζ₀ and has a growth rate of f_ζ0(n).
Etation is defined as {a,b[1\1\1\2]2}. It has the equivalent ordinal of η₀ and has a growth rate of f_η0(n).
Gammation is defined as {a,b[1/2]2}. It has the equivalent ordinal of Γ₀ and a growth rate of f_Γ0(n), or f_φ(1,0,0)(n) using the extended Veblen function.

Section 5: Extrapolated Names for Functions Past f_ε0(n)

CONTENT WARNING: Cursed ordinals ahead! If ill-defined ordinal arithmetic makes you feel ill please step away now.

We're going to assume from this point on that ordinal tetration and higher ordinal hyperoperators just work and are not ill-defined. If we do so, it is easy to continue the pattern of expomegation and tetramegation with pentamegation, which would be defined as {a,b[[1]]2} = {a,b[1\2]2} and have a growth rate of f_ε0(n). You could say that ε₀ has a pentational factor if you wish. Following this, duopentamegation is be defined as {a,b[1\1[1\2]2]2} and has a growth rate of f_εε0(n). Hexamegation is defined as {a,b[1\1\2]2} and has a growth rate of f_ζ0(n).

Funny thing, we're running into the exact issue that started this whole naming scheme. No problem, we can just plug our naming scheme into itself, giving us megamegation. Megamegation is defined as {a,b[1[2]\2]2} = {a,b[1\\2]2} and has a growth rate of f_φ(ω,0)(n) using the Veblen function. To continue this pattern, it would be wise to "simplify" our ordinals into a more understandable (yet ill-defined) pattern. if we replace ε₀ with ω↑↑ω = ω↑↑↑2 and replace ζ₀ with ω↑↑↑ω = ω{4}2, we could say that φ(ω,0) = ω{ω}ω = {ω,ω,ω}. In fact, if we define φ(ω,0) as being equal to f_ω(ω), we can plug FGH into itself and create f_fω(ω)(n) as a substitute for f_φ(ω,0)(n). Doing so makes calling our function name "megamegation" more clear. And now that it's clarified we can continue further.

Duomegamegation has a growth rate of f_fω2(ω)(n), which I believe would be equal to f_φ(ω2,0)(n). If so, it would be defined as {a,b[1\\1\\2]2}.
Expomegamegation has a growth rate of f_fω²(ω)(n), which I believe would be equal to f_φ(ω²,0)(n). If so, it would be defined as {a,b[1[3]\2]2}.
Tetramegamegation has a growth rate of f_fω^ω(ω)(n), which I believe would be equal to f_φ(ω^ω,0)(n). If so, it would be defined as {a,b[1[1,2]\2]2}.
And finally, megamegamegation has a growth rate of f_{fφ(ω,0)(ω)}(n), which I believe would be equal to f_{φ(φ(ω,0),0)}(n). If so, it would be defined as {a,b[1[1\\2]\2]2}.

And I think that's a good place to stop.